Incompressible surfaces and pseudo-Anosov flows
A dichotomoy exists between \(\pi_1\)-injective maps \(g\) of a closed surface \(S\) into a hyperbolic 3-manifold \(M\): by results of Bonahon, Marden and Thurston, either there is a finite covering space \(\widehat M\rightarrow M\) and a lift \(\widehat g:S\rightarrow\widehat M\) which is the inclusion of a fiber of a fibration \(\widehat M\rightarrow S^1\) or \(g_*\pi_1(S)\subset\pi_1(M)\) is quasi-Fuchsian. In the first case \(g_*\pi_1(S)\) is a geometrically infinitely discrete group of isometries of hyperbolic space and the second case \(g_*\pi_1(S)\) is geometrically finite. The present paper gives criteria for the geometrically infinite case in terms of a quasi-geodesic pseudo-Anosov flow \(\Phi\) on \(M\) which satisfies two additional global conditions. The map \(g\) is assumed to be an immersion transverse to \(\Phi\). Then \(g_*\pi_1(S)\) is geometrically infinite if and only if every lift \(\widetilde g:\widetilde S\rightarrow\widetilde M\) meets every orbit of the lifted flow on \(\widetilde M\). In that case \(g\) lifts to an embedded fiber and \(\Phi\) lifts to a suspension flow. The paper is clearly and carefully written. The methods of proof involve work of Fried and Mosher on cross-sections to pseudo-Anosov flows, and work of Gabai and Thurston on the norm of homology of 3-manifolds. Several interesting dynamical properties of these flows are discussed as well.
scientific article; zbMATH DE number 1307965